Method for image coding by rate-distortion adaptive zerotree-based residual vector quantization and system for effecting same

ABSTRACT

Methods and systems for encoding, transmitting and decoding digital images by rate-distortion adaptive zerotree-based residual vector quantization are disclosed. A method of the invention includes receiving a digital image, transforming the digital image into the wavelet domain generating a pyramid hierarchy, losslessly encoding a top LL subband from the pyramid hierarchy, encoding other subbands by vector quantization based on a zerotree insignificance prediction, generating an encoded image from the lossless encoding and vector quantization encoding, transmitting the encoded image along a communications channel, receiving the encoded image transmitted along the communications channel, reconstructing a zerotree from the encoded image, vector quantization decoding subbands from the encoded image other than a top LL subband, losslessly decoding the top LL subband from the encoded image, reverse wavelet transforming the top LL subband and the vector quantization decoded subbands and outputting a decoded image.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of application Ser. No. 09/739,168,filed Dec. 18, 2000, now U.S. Pat. No. 6,944,350, issued Sep. 13, 2005,which claims benefit of priority pursuant to 35 U.S.C. § 119(e), fromprovisional patent application Ser. No. 60/172,708, filed Dec. 17, 1999,both of which are incorporated by reference.

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention relates to the field of image compression. Moreparticularly, this invention relates to methods and systems forefficiently compressing still images and video frames using wavelettransformation and vector quantization.

2. State of the Art

Image compression may be classified into two categories of encoding:lossless and lossy. Lossless encoding techniques guarantee that thedecompressed image from the compressed (encoded) data is identical tothe original image. Lossy encoding is generally capable of achievinghigher compression ratios versus lossless encoding, but at the expenseof some loss of image fidelity. Exemplary conventional lossless imageencoding techniques include run-length encoding, Huffman encoding, andLempel/Ziv encoding. Exemplary conventional lossy image encodingtechniques include transform encoding, vector quantization (VQ),segmentation and approximation methods, spline approximation methods andfractal encoding.

Image compression is particularly useful for transmitting and displayinggraphical images over the Internet, where it takes more time to transmitthe original image than a compressed image. Image compression is alsouseful for compressing digital video frames. The compression of digitalvideo image frames is particular useful for such applications as videoconferencing and video streaming.

The basic idea behind transform coding is decorrelating the originalsignal so that the signal energy may be redistributed among only a smallset of transform coefficients. In this way, many coefficients may bediscarded after quantization and before encoding. Generally, transformcoding involves four steps: (1) image subdivision, which divides animage into smaller blocks, (2) transformation, such as a Discrete CosineTransform (DCT) and a wavelet transform, (3) quantization, such as zonalcoding and thresholding, and (4) encoding, such as Huffman encoding.

With regard to the quantization, vector quantization is an alternativeto scalar quantization and may lead to better performance according toP.I.R.A. International, “Open information interchange study onimage/graphics standards,” Tech. Rep. Appendix, PIRA International, June1993. A vector quantizer is a system which maps a K-dimensionalEuclidean space R^(k) to a finite subset X in R^(k) made up of Nvectors. This subset X becomes the vector codebook. An image can then berepresented by the index of the codebook, and thus, be compressed.

Wavelet transforms have been applied to image compression. See e.g., M.Anotonini et al., “Image coding using wavelet transform,” IEEETransactions on Image Processing, vol. 1, no. 2, pp. 205-220; J. M.Shapiro, “Embedded image coding using zerotree of wavelet coefficients,”IEEE Transactions on Signal Processing, vol. 41, pp. 3445-3462, December1993; D. Sampson et al., “Wavelet transform image coding using latticevector quantization,” Electronics Letters, vol. 30, pp. 1477-78,September 1994; and P. C. Cossman et al., “Tree-structured vectorquantization with significance map for wavelet image coding,” Proc. 1995IEEE Data Compression Conf (DCC), March 1995.

Wavelets are mathematic functions that provide joint time-frequencyrepresentation of a signal. Wavelets decompose data into differentfrequency components (known as subbands) and each component can betreated with a resolution matched to its scale. Wavelet transforms withthe feature of joint locality can generate “sparse” coefficients, whichare particularly useful for image compression. Additionally, the pyramidhierarchy of wavelet decomposition also enables many compressionalgorithms based on inter-band and cross-band relationships, such aszerotrees. See e.g., Shapiro supra. Although the wavelet transformreduces the correlation between image samples, high-order statisticaldependencies still exist within or across subband coefficients. A vectorquantizer may exploit these high-order statistical dependencies byjointly quantizing several coefficients. See e.g., A. N. Akansu et al.,Subband and Wavelet Transforms: Design and Applications, Kluwer AcademicPublishers, Norwell, Mass., 1996.

A wavelet is a mathematical function that satisfies certain mathematicalrequirements and is used in representing data (signal) or otherfunctions. A wavelet provides an efficient and informative descriptionof a signal, and is superior to the traditional Fourier transform inmany fields, especially for fast transient and non-stationary signals.The wavelet has many useful properties, such as joint time(spatial)-frequency localization and multi-resolution representation.

Basis functions for the Fourier transform are sinusoids. In contrast,with the wavelet transform, various basis functions may be designedbased on the features of a specific application. Most wavelets do nothave analytical solutions. The wavelet transform may be implemented byiterating the quadrature mirror filters in a tree algorithm, as known toone of ordinary skill in the art. However, the wavelet tree algorithm asdisclosed in Y. Sheng, The Transforms and Applications Handbook, Chapter“Wavelet Transform,” CRC Press, in cooperation with IEEE, 1996, permitsfast wavelet transform, and only requires fewer operations that theconventional fast Fourier transform (FFT).

In most cases images are stored and transmitted by integer or binaryformat. For hardware implementation of image processing, pure integeroperation is often preferred. However, most filters, such as waveletsand wavelet packets, have floating point coefficients. Thus, it isdesirable to use an efficient integer implementation of wavelettransform for image coding.

The lifting scheme as disclosed in I. Debauchies et al., “Factoringwavelet transform into lifting steps,” Tech. Rep. Lucent, BellLaboratories, 1996, the disclosure of which is incorporated herein byreference in its entirety for all purposes, supports perfectreconstruction and fast computation. In the lifting scheme, the wavelettransform is performed in the spatial domain. The basic idea behind thelifting scheme is a predict-update procedure, where the prediction erroris related to the high-pass band and the updated prediction is relatedto the low-pass band.

With the proliferation of digital imagery in many applications includingthe Internet, there is a need in the art for methods and systems thatperform image compression and decompression (coding and decoding) usinga combination of lossless and lossy image encoding and decoding toobtain a high peak signal-to-noise ratio (PSNR) and high rates of speed.

BRIEF SUMMARY OF THE INVENTION

The present invention includes methods for image encoding byrate-distortion adaptive zerotree-based residual vector quantization andsystems for effecting same.

A method embodiment for encoding a digital image by rate-distortionadaptive zerotree-based residual vector quantization in accordance withthe present invention includes: obtaining a digital image, transformingthe digital image into wavelet domain, thereby generating a pyramidhierarchy, losslessly encoding a top low-low (LL) subband of the pyramidhierarchy, thereby obtaining a losslessly encoded portion of the digitalimage, vector quantization (VQ) encoding all other subbands of thepyramid hierarchy based on a zerotree insignificance prediction therebyobtaining a lossy encoded portion of the digital image, and outputtingan encoded image from the losslessly encoded portion and the lossyencoded portion of the digital image.

A method embodiment for decoding an image encoded by rate-distortionadaptive zerotree-based residual vector quantization in accordance withthe present invention includes: obtaining an encoded image,reconstructing a zerotree from the encoded image, vector quantizationdecoding subbands in the encoded image other than a top LL subband,losslessly decoding the top LL subband, reverse wavelet transforming thetop LL subband and the vector quantization decoded subbands; andoutputting a decoded image from the decoded top LL subband and thedecoded subbands other than the decoded top LL subband.

An integrated circuit embodiments for implementing a method for encodingand decoding, respectively, a digital image by rate-distortion adaptivezerotree-based residual vector quantization in accordance with thepresent invention are disclosed.

An integrated circuit embodiment for coding and decoding an image byrate-distortion adaptive zerotree-based residual vector quantization inaccordance with the present invention is disclosed.

A circuit card embodiment for implementing a method for encoding anddecoding an image using rate-distortion adaptive zerotree-based residualvector quantization in accordance with the present invention isdisclosed.

A system embodiment for encoding, transmitting and decoding an imageusing rate-distortion adaptive zerotree-based residual vectorquantization in accordance with the present invention is disclosed.

These embodiments and methods of the present invention will be readilyunderstood by reading the following detailed description in conjunctionwith the accompanying figures of the drawings.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

In the drawings, which illustrate what is currently regarded as the bestmode for carrying out the invention and in which like reference numeralsrefer to like parts in different views or embodiments:

FIG. 1 is a flow chart of a method of encoding a digital image inaccordance with the present invention.

FIG. 2 is a block diagram of a DPCM encoder and a DPCM decoder inaccordance with the method of the present invention.

FIG. 3 is an illustration of a predictor based on a three-pointoperation in accordance with the method of the present invention.

FIG. 4 is a block diagram of Universal source coder in accordance withthe method of the present invention.

FIG. 5 is a diagram illustrating imagetree and threshtree in a pyramidhierarchy in accordance with the method of the present invention.

FIG. 6 is a tree diagram illustrating the structure of imagetree andthreshtree according to the present invention.

FIG. 7 is a method for constructing a significance map of a threshtreein accordance with the present invention.

FIG. 8 is a block diagram of the method of encoding and decoding imagesin accordance with the invention.

FIG. 9 are block diagrams of integrated circuits implementing the methodof encoding and decoding of still images in accordance with the methodof FIG. 8.

FIG. 10 is a block diagram of a circuit card embodiment of the inventionincluding the method disclosed in FIG. 8.

FIG. 11 is a block diagram of a system embodiment of the inventionincluding the method disclosed in FIG. 8.

DETAILED DESCRIPTION OF THE INVENTION

The invention is a method for image coding by rate-distortion adaptivezerotree-based residual vector quantization and system for effectingsame. While the method of the invention has many applications, themethod of the invention is particularly useful for compressing imagesfor transmission through a communication channel and then reconstructingthe images at a remote location. The method of the invention uses adiscrete integer wavelet transform using the lifting scheme. The terms“discrete integer wavelet transform,” “discrete wavelet transform” and“DWT” are used interchangeably herein.

The wavelet bases of the method were selected based on the followingconsiderations. First, perfect reconstruction is desirable for imagecoding, transmission and decoding. Second, since most images are smooth,it is desirable to use those mother wavelets with reasonably highvanishing moments. Additionally, the length of the finite impulseresponse (FIR) filters should be short so to enable fast computation andedge treatment. Note that for octave decomposition, the size of coarsersubband becomes quite small when the decomposition level increases.Furthermore, it is desirable to have FIR filters that are linear phase,since that allows a cascaded pyramid structure without phasecompensation. Qun Gu, Image Coding by Rate-Distortion AdaptiveZerotree-based Residual Vector Quantization (2000) (published Master ofScience (M.S.) thesis, Utah State University, Logan, Utah, on file withthe Utah State University Library), details methods of encoding anddecoding images in accordance with the present invention, the entirecontents of which are incorporated herein by reference for all purposes.

FIG. 1 is a flow chart of a method 100 of encoding a digital image inaccordance with the present invention. Method 100 may include obtaining102 a digital image, transforming 104 the digital image into waveletdomain, thereby generating a pyramid hierarchy, losslessly encoding 106a top low-low (LL) subband of the pyramid hierarchy, thereby obtaining alosslessly encoded portion of the digital image, vector quantization(VQ) encoding 108 all other subbands of the pyramid hierarchy based on azerotree insignificance prediction, thereby obtaining a lossy encodedportion of the digital image, and outputting 110 an encoded image fromthe losslessly encoded portion of the image and the lossy encodedportion of the digital image.

Transforming 104 the digital image into wavelet domain may beaccomplished using a 2-dimensional (2-D) separable octave decompositionwhich generates the pyramid hierarchy. Losslessly encoding 106 a top LLsubband of the pyramid hierarchy may be accomplished using, for exampleand not by way of limitation, a differential pulse coded modulator(DPCM) in combination with Huffman coding, DPCM in combination withUniversal source coding, DPCM in combination with arithmetic coding, andother suitable lossless encoding techniques known to one of ordinaryskill in the art.

Wavelet transforms suitable for performing the discrete integer wavelettransform 104 according to the method of the invention may include, forexample and not by way of limitation, the Daubechies' 9-7 symmetricwavelet transform, the Two Six (TS) transform, and the Two Ten (TT)wavelet transform.

The Daubechies' 9-7 symmetric biorthogonal wavelet transform is thepresently preferred wavelet because it provides a good trade-off amongthe above considerations. The symmetric property of the Daubechies' 9-7wavelet allows simple edge treatment. The analysis low-pass (LP) filterof the Daubechies' 9-7 wavelet of the present invention has nine tapsand the analysis high-pass (HP) filter of the Daubechies' 9-7 wavelet ofthe present invention has seven taps. Both analysis and synthesishigh-pass filters have four vanishing moments. This implies that thetransform coefficients will be zero (or close to zero) for any signalthat can be described by (or approximated by) a polynomial of 4th orderor less. The filter coefficients of the Daubechies' 9-7 filter pair areshown in Table 1 below.

TABLE 1 n 0 ±1 ±2 ±3 ±4 2^((−1/2))h_(n) 0.602949 0.266864 −0.078223−0.016864 0.026749 2^((−1/2)){tilde over (h)}_(n) 0.557543 0.295636−0.028772 −0.045636 0

As noted above, the Daubechies' 9-7 symmetric wavelet transform of thepresent invention is implemented by the lifting scheme. The waveletcoefficients derived from the lifting scheme for the Daubechies' 9-7symmetric wavelet transform are given by equations (1)-(4):d _(1,l) ⁽¹⁾ =s _(0,2l+1)+└α(s _(0,sl) +s _(0,2l+2))+½┘  (1)s _(1,l) ⁽¹⁾ =s _(0,2l)+└β(d _(1,l) ⁽¹⁾ +d _(1,l−1) ⁽¹⁾)+½┘  (2)d _(1,l) =d _(1,l) ⁽¹⁾+└γ(s _(1,l) ⁽¹⁾ +s _(1,l+1) ⁽¹⁾)+½┘  (3)s _(1,l) =s _(1,l) ⁽¹⁾+└δ(d _(1,l) +d _(1,l−1))+½┘  (4)where “d” represent HP coefficients and “s” represent LP coefficientsand where “└ ┘” denotes truncation for integer operations. The constantsderived from the Daubechies' 9-7 symmetric wavelet transform filter pair(see Table 2) and the lifting scheme may be given by the followingapproximations:α≈−1.586134342β≈−0.05298011854γ≈−0.8829110762δ≈−0.4435068522K≈−1.149604398  (5)

The TS transform of the present invention is an integer wavelettransform derived from the (3, 1) biorthogonal wavelet transform ofCohen-Daubechies-Feauveau as disclosed in A. Cohen et al., “Biorthogonalbases of compactly supported wavelets,” Comm. Pure Appl. Math., vol. 45,pp. 485-560, 1992, the disclosure of which is expressly incorporatedherein by reference in its entirety for all purposes. Here, the notation(3, 1) refers to the vanishing moments of the analysis and synthesis HPfilter separately. The TS transform of the present invention has a twotap LP analysis filter, and a six tap HP analysis filter.

Implementation of the TS transform by the lifting scheme according tothe method of the present invention includes calculating the followingwavelet coefficients:

$\begin{matrix}{d_{1,l}^{(1)} = {s_{0,{{2l} + 1}} - s_{0,{2l}}}} & (6) \\{s_{1,l} = {s_{0,{2l}} + \left\lfloor \frac{d_{1,l}^{(1)}}{2} \right\rfloor}} & (7) \\{d_{1,l} = {d_{1,l}^{(1)} + \left\lfloor {\frac{s_{1,{l - 1}}}{4} - \frac{s_{1,{l + 1}}}{4} + \frac{1}{2}} \right\rfloor}} & (8)\end{matrix}$

It is known that even though the TS transform has three vanishingmoments, it generally performs worse than those filters with comparablenumber of analysis vanishing moments. See for example, A. R. Calderbanket al., “Wavelet transforms that map integers to integers,” MathematicsSubject Classification, August 1996. For this reason the Daubechies' 9-7transform is preferred over the TS transform.

The TT transform of the present invention is derived from theCompression with Reversible Embedded Wavelets (CREW) system disclosed inM. J. Gormish et al., “Lossless and nearly lossless compression for highquality images,” Tech. Rep., Ricoh California Research Center, 2882 SandHill Road, Suite 115, Menlo Park, Calif. 94025-7022, the contents ofwhich are expressly incorporated herein by reference in its entirety forall purposes. The TT transform of the present invention has two taps inthe LP analysis filter and ten taps in the HP analysis filter. Thefilter pair of the TT transform of the present invention is derived fromthe LeGall-Tabatabai polynomial of case “p=3,” which is of degree 10 inz⁻¹. The TT transform of the present invention is defined by:

$\begin{matrix}{{s(n)} = \left\lfloor \frac{{x\left( {2n} \right)} + {x\left( {{2n} + 1} \right)}}{2} \right\rfloor} & (9) \\{{d(n)} = {{x\left( {2n} \right)} - {x\left( {{2n} + 1} \right)} + {p(n)}}} & (10)\end{matrix}$where p(n) is defined by:

$\begin{matrix}{{p(n)} = \left\lfloor \frac{{3{s\left( {n - 2} \right)}} - {22{s\left( {n - 1} \right)}} + {22{s\left( {n + 1} \right)}} - {3{s({n2})}} + 32}{2} \right\rfloor} & (11)\end{matrix}$The TT transform is considered one of the most efficient waveletdecompositions. The special properties of the TT transform include: (1)the LP (smooth) coefficient has no growth in bit depth, i.e., if theinput signal can be represented by b bits, the output LP coefficient canbe represented by b bits, and (2) the HP (detailed) output coefficientsrequire two additional bits for perfect reconstruction. Thus, the TTtransform of the present invention is exactly reversible.

Because biorthogonal wavelet transformations in most cases cannotpreserve energy according to Parseval's relation, it is necessary toemploy a weighting scheme in order to use rate-distortion optimizationtechniques based on orthogonal transforms. When Parseval's relationholds, the distortion in the wavelet domain can be directly related tothe distortion in the image domain. It is known that the biorthogonalwavelet transform with a scalar weighting scheme may result in minimumreconstruction error when using rate-distortion optimization techniquesbased on orthogonal transforms.

Let h(n) and g(n) denote the synthesis filter pair of a biorthogonalwavelet transform, and w_(l) and w_(h) as the weights for the LP and HPtransform coefficients respectively. When the wavelet transform isimplemented by the lifting scheme, the weights can be derived as:

$\begin{matrix}{w_{l} = \frac{\sqrt{2}{\sum\limits_{L}{h(n)}^{2}}}{{{\sum\limits_{L}{\left( {- 1} \right)^{n}{h(n)}}}}^{2}}} & (12) \\{w_{h} = \frac{\sqrt{2}{\sum\limits_{L}{g(n)}^{2}}}{{{\sum\limits_{L}{\left( {- 1} \right)^{n}{h(n)}}}}^{2}}} & (13)\end{matrix}$where L is the length of the filter. The weights of a two-dimensional(2-D) multilevel wavelet transform can be obtained by straight forwardextension of equations (12) and (13).

Generally, most of the energy of images remains in the low-frequencyband. After wavelet decomposition, the top LL subband contains mostlythe low-frequency coefficients and corresponds to the whole image withfewest coefficients compared with all other subbands. Therefore, the topLL subband contributes most to the quality of the reconstructed image.For this reason, the top LL subband is coded losslessly (see 106,FIG. 1) in the method 100 of encoding a digital image according to thepresent invention. Since the statistical distribution of the top LLsubband is similar to the original image, it is reasonable to use adifferential pulse coded modulator (DPCM) technique to encode it. Theprediction error by DPCM is then less correlated and may be coded bysome entropy coding methods. In the method 100 of encoding a digitalimage according to the present invention, lossless coding methods mayinclude, for example and not by way of limitation, DPCM plus Huffmancoding, DPCM in combination with Universal source coding or Rice coding,DPCM and arithmetic coding and any other suitable lossless coding methodknown to one of ordinary skill in the art.

DPCM is a well known technique for compression of highly correlatedsource signals, e.g., speech and images. In channel coding, thedifferential nature of DPCM makes it sensitive to bit error. In sourcecoding, DPCM generally can compress images to three or four bits perpixel with minimal distortion. A conventional form of a DPCM encoder 200and DPCM decoder 250 are shown in FIG. 2.

In traditional terms, it has been said that DPCM is used to remove the“redundancy” of the source signal. Strictly speaking, DPCM is used toremove the statistical correlation between samples of the source signal.In the 2-D case, such as image processing, the predictor 220 may be a2-D linear causal predictor which may be expressed as:

$\begin{matrix}{{\overset{\sim}{u}\left( {m,n} \right)} = {\sum\limits_{k,{l \in}}{\sum\limits_{W}{{a\left( {k,l} \right)}{\overset{\sim}{u}\left( {{m - k},{n - l}} \right)}}}}} & (14)\end{matrix}$where W is a causal prediction window. In the method 100 according tothe present invention, the predictor 220 may be based on a three-pointoperation:ũ(m,n)=ũ(m,n−1)+ũ(m−1,n)+ũ(m−1,n)+ũ(m−1,n−1)  (15)and illustrated in FIG. 3.

The prediction error generated by a DPCM scheme is then coded by aHuffman coding technique. See for example, A. K. Jain, Fundamentals ofDigital Image Processing, Prentice Hall, Englewood Cliffs, N.J. 07632,1989, the disclosure of which is incorporated herein by reference in itsentirety for all purposes. Huffman coding is among the best known andwidely used entropy coding techniques. Huffman coding is a variablelength coding based on the probability distribution of the sourcesymbols. The basic idea of Huffman coding is to assign shorter codewordsto frequently occurring, more probable symbols, and assign longercodewords to infrequently occurring, less-probable symbols. Huffmancoding can achieve the optimal performance, or the compressed bit-ratethat is equal to the entropy of the source symbols, only when theprobabilities of the source symbols happen to be negative powers of 2.However, this case is rare in practice.

Huffman coding is known to be often quite effective for a predictivecoding method. However, because Huffman coding is non-adaptive and basedon the statistical distribution of a set of images, it is difficult tobuild a Huffman table which can give best performance for eachindividual image. For this reason, an adaptive coding method, such asUniversal source coding may also be used to losslessly encode 106 thetop LL subband of the pyramid hierarchy.

The basic idea behind Universal source coding is to first executemultiple lossless coders in parallel and then select the one that canproduce the fewest bits for a given period of time, then send a smallamount of overhead to inform the decoder which coder was selected. Aparticular Universal source coding method which may be used tolosslessly encode 106 a top LL subband of the pyramid hierarchy inaccordance with the present invention is disclosed in R. F. Rice, “SomePractical Universal Noiseless Coding Techniques, part iii, module psi14,k+,” Tech. Rep., Jet Propulsion Laboratory, 1991, the disclosure ofwhich is incorporated herein by reference in its entirety for allpurposes and referred to herein as “Rice coding.”

A block diagram of Universal source coder 400 is shown in FIG. 4. Thedata is input in blocks rather than by symbol and the output is likewisein block form. For the case of image data, the block can be a 2-D vectorof the image to be coded. The reversible preprocessor 410, may be a DPCMcoder with a 2-D predictor similar to that used in DPCM plus Huffmancoding described above. The standard source which is output from thereversible preprocessor 410 is defined as a set of nonnegative integersamples, whose probability is inversely related to its value. That is,for a sequence of J samples, the standard source is:{tilde over (δ)}^(n)=δ₁δ₂ . . . δ_(J)  (16)where the J samples of the standard source are of values from a set ofnonnegative integers, 0, 1, 2, . . . , q−1, and the samples of thestandard source have the probability distribution:P _(δ) ={p ₀ , p ₁ , . . . , p _(q−1)}  (17)If the samples of the standard source are independent of themselves andany side information, and the order of their probability distributionsatisfies:p₀≧p₁≧ . . . ≧p_(q−1)  (18)then the standard source is known as an idealized standard source. Thus,an idealized standard source is a set of symbols sorted in the order oftheir probability distribution, and is ready to build a Huffman table.

The prediction error generated by the reversible preprocessor 410 tendsto be unimodal distributed, and thus, can well approximate a standardsource. The prediction error is mapped into nonnegative values by:θ({tilde over (x)} _(j))=min({tilde over (x)} _(j), 2^(n)−1−{tilde over(x)} _(j))  (19)For n-bit data which produces n+1 bit prediction errors, this mappingcan constrain them to still only n bits. This approximate standardsource is input to the adaptive variable length coder 420, e.g., aHuffman coder. The adaptive variable length coder 420 includes a set ofsubcoders (not shown). Each subcoder gives best performance at a certaindisjoint entropy range. The bits of an N-bit input sample are firstspilt into two parts: N-k most significant bits and k least significantbits. The sequence of k least significant bits is randomly distributedand does not require entropy coding. The N-k most significant bits arecoded by a fixed Huffman coder. All the subcoders run in parallel. Thebest result is selected, and the index of this subcoder is sent as sideinformation to inform the decoder.

Several parameters are required to configure a Universal source coderfor losslessly encoding 106 a top LL subband of the pyramid hierarchy:(1) number of bits per sample—the number of bits per sample actuallydepends on the dynamic range of the input data, (2) number ofsubcoders—generally, incrementing the index of a subcoder will shift itsdynamic range (entropy range) by 1 bit per sample, thus, it will bereasonable to set N subcoders for an N-bit input sequence, and (3) blocksize, J—it is recommended that the block size, J, be chosen in the range1≦J≦16, and most applications work well with a fixed J=16.

An important aspect of the method 100 of the present invention is thezerotree quantizer on the discrete wavelet transform (DWT) coefficientsperformed in the VQ encoding 108 of subbands other than the LL subband,hereinafter referred to as the wavelet rate-distortion adaptive residualvector quantizer (WRDADRVQ). The zerotree prediction is implicitlyapplied in the rate-distortion optimization. No hard thresholding isrequired according to the method 100 of the present invention.

In the octave pyramid decomposition, the HL, HH, LH subbands correspondto the horizontal, diagonal, and vertical directions, respectively. Eachdecomposition level is related to a different resolution scale. Thus, itis reasonable to use different vector shapes and sizes for differentdecomposition levels and subbands. However, irregular vector sizes andshapes may complicate the zerotree hierarchy and may reduce thecapability of the insignificance prediction. In order to preserve thezerotree prediction, and for simplicity, uniform square vectors for allsubbands are used in the method 100 of the present invention. For theWRDADRVQ, each vector in the top LL subband has three direct childrenvectors, and each of these three direct children vectors has fourchildren vectors, and each of these four children vectors has fourchildren vectors at the next level, recursively.

The parent-descendent relationship constructs a tree structure, whichconsists of a root vector at the top LL subband, and the root vector hasthree branches in the three directions (horizontal, diagonal andvertical), and each branch itself is a tree that exponentially grows bya power of four. This tree structure starting from the top LL rootvector or subband is referred to herein as an “imagetree.” The treestructure of each branch is referred to herein as a “threshtree.” Thus,each imagetree has three threshtrees in the three directions. FIG. 5illustrates an imagetree and its threshtrees in a pyramid hierarchy.Note that the root of an imagetree, located in the top LL subband, islosslessly coded in accordance with the method of the present invention.However, for both encoding and decoding, the top L subband is scannedvector by vector in the order of row by row. Since each vector in thetop L subband is an imagetree root and identifies an imagetree, when anyvector in the top LL subband is scanned, all of its three threshtreeswill be coded by VQ with a rate-distortion optimization within thethreshtree. In this way, after scanning the top LL subband, all thevectors in the image except for the top LL subband will be coded by VQ(see 108 of FIG. 1).

When a vector is coded by an adaptive rate-distortion optimal VQ, theresulting number of VQ stages and the indices of each coding unit willbe converted into bit-stream and sent to the communication channel orsaved into a data file. The bit format of a coded vector according tothe present invention is shown in Table 2.

TABLE 2 VQ Header [VQ index 0] [VQ index 1] . . . [VQ index n]The first (leftmost) portion of the bit format is the VQ header, whichis the actual VQ stage number coded by the Huffman table, then followedby the indices of each VQ coding unit, if any. The VQ indices aredirectly packed into bit-stream without further coding. In the methodaccording to the present invention, the number of vectors per codingunit is constant. So, the bit length of the VQ index of each coding unitis also constant.

The goal of rate-distortion optimization within a vector for an N-codingunit adaptive rate-distortion optimal VQ is to find the optimal numberof the VQ coding unit, n, which gives the minimal cost J:

$\begin{matrix}{J = {\min\limits_{N}\left( {{D(n)} + {\lambda\;{R(n)}}} \right)}} & (20)\end{matrix}$where D(n) is the distortion of n-coding unit VQ and

$\begin{matrix}{{R(n)} = {{H(n)} + {\sum\limits_{i = 1}^{n}{B(i)}}}} & (21)\end{matrix}$is the bit length of coded VQ. H( )is the length of the entropy codeused to code the VQ header, and B(i) is the bits per coding unit for VQindex and is a constant in accordance with the present invention. Thevalue of the VQ header, h, may be the number of the coding units for theadaptive rate-distortion optimal VQ. Since the number of maximum VQcoding units, N, is usually small, this optimization may be implementedby a simple linear search as known to one of ordinary skill in the art.

The following notation is used herein to denote the significance mapsymbols in the method of the present invention: SG—significant,RT—zerotree root, IZ—isolated zero, and CH—zerotree child. Only thosesignificant vectors are actually coded by VQ. For RT and IZ, only thesignificance map symbols are sent. Zerotree children do not need to becoded because they can be predicted by their zerotree root. Thesignificance map symbols are embedded in the VQ header. Hence, thevalue, h, of the VQ header may be defined as follows:

$\begin{matrix}{h = \left\{ \begin{matrix}0 & {{if}\mspace{14mu}{IZ}} \\{\left. 1 \right.\sim N} & {{if}\mspace{14mu}{SG}} \\{N + 1} & {{if}\mspace{14mu}{RT}}\end{matrix} \right.} & (22)\end{matrix}$

To illustrate the rate-distortion optimization along a threshtree,denote the function T(s), which is defined as:

$\begin{matrix}{{T(s)} = \left\{ \begin{matrix}J & {{{for}\mspace{14mu} s} = {SG}} \\{{D(0)} + {\lambda\;{H\left( {N + 1} \right)}}} & {{{for}\mspace{14mu} s} = {RT}} \\{{D(0)} + {\lambda\;{H(0)}}} & {{{for}\mspace{14mu} s} = {IZ}} \\{D(0)} & {{{for}\mspace{14mu} s} = {CH}}\end{matrix} \right.} & (23)\end{matrix}$where D(0) is the distortion of the vector without VQ coding (or the“zero coding unit”). If mean square error is chosen as the measure ofdistortion, then D(0) is the average energy (or power) of the vector.The value of the function T(s) is the actual cost of a vector subject toits significance. Let W_(l,p) be the total cost of the pth node at l DWTlevel and the costs of all its descendants, then W_(l,p) can beexpressed recursively as:

$\begin{matrix}{W_{l,p} = {{T\left( s_{({l,p})} \right)} + {\sum\limits_{q = {4p}}^{q < {{4p} + 4}}W_{{l - 1},q}}}} & (24)\end{matrix}$where q is the index of the four siblings belonging to a same parent.Note that the DWT level is defined as in FIG. 6, which illustrates thestructure of imagetree and threshtree according to the presentinvention. Also note that l DWT level consists of 4^(l) nodes, startingfrom Level 0, as 1, 4, 16, . . . . For an L level DWT decomposition, thevery bottom level satisfies:W _(L−1,p) =T(s _((L−1,p)))  (25)The minimal W_(0,0) may be found by varying the s_((1,p)), or in otherwords, by a proper zerotree classification. The significance map of thethreshtree is constructed from the bottom (finest) level to the top(coarsest) level (Level 0 in FIG. 6).

Referring to FIG. 7, a method 700 for constructing a significance map ofa threshtree in accordance with the present invention is shown. Method700 includes searching 702 for the optimal VQ coding unit number, n, forall the vectors of the threshtree by minimizing the rate-distortion costof each individual vector, recoding the minimal distortions, the optimalVQ coding unit, the resulting VQ indices, and letting l=L−1, assigning704 the value zero top, computing 706 the minimum total cost of the pthzerotree node at l DWT level and the cost of all its descendants,incrementing 708 the value of p, if the inequality 710, p<4^(l), istrue, then returning to 706, otherwise if the equality 712, l=0, is truethen stopping, otherwise decrementing 714 the value of l and returningto 704.

Computing 706 the minimum total cost of the pth zerotree node at l DWTlevel and the cost of all its descendants, i.e., min_(s(1,p))(W_(l,p)),involves the following rules when comparing the three cases ofs_((l,p)): SG, IZ, RT.

If s_((l,p))=SG, then

$\begin{matrix}{W_{l,p} = {{T\;({SG})} + {\sum\limits_{q = {4p}}^{q < {{4p} + 4}}\; W_{{l - 1},q}}}} & (26)\end{matrix}$with the significance of all its descendants unchanged, i.e., the secondpart of equation (24) unchanged. If s_((l,p))=IZ, then

$\begin{matrix}{W_{l,p} = {{T\;({IZ})} + {\sum\limits_{q = {4p}}^{q < {{4p} + 4}}\; W_{{l - 1},q}}}} & (27)\end{matrix}$with the significance of all its descendants unchanged. If s_((l,p))=RT,then

$\begin{matrix}{W_{l,p} = {{T\;({RT})} + {\sum\limits_{q = {4p}}^{q < {{4p} + 4}}\; W_{{l - 1},q}}}} & (28)\end{matrix}$by changing the significance of all its descendants as CH. Find theminimal cost among these three cases, i.e., equations (26)-(28). Ifequation (28) is minimal among all three, update W for all itsdescendants.

The WRDADRVQ scheme does not support embedded or successive coding. Therate control is accomplished by applying the proper λ value in equation(23). For example and not by way of limitation, after the wavelettransform 104, lossless encoding 106 and VQ search 702, all VQ indicesand the distortions of each coding unit are bookmarked. The wavelettransform and VQ search, which contribute most of the computationalcomplexity, are obtained only once in the method 700. Then the λ valuecorresponding to the targeted rate may be performed by general searchingmethods, such as bi-section, golden search or other suitable methodsknown to one of ordinary skill in the art. The rate output by methodsteps 704-714 may be used to compute a fitness function.

Decoding an image encoded according to the method 100 of the presentinvention includes decoding the top LL subband by the lossless decodercorresponding to the lossless encoder. Since the bit-stream of VQ codeddata is in the order of imagetree by imagetree, the pyramid hierarchymay be reconstructed by putting back the coefficients of the imagetreeto their corresponding positions. If any vectors are predicted aszerotree children, the decoder slips the lower levels of the threshtreeand fills the corresponding vectors with zero. The decoding is completedby performing an inverse DWT on the reconstructed pyramid hierarchy andthe reconstructed image is complete. The method of decoding according tothe present invention may include obtaining the encoded image,reconstructing a zerotree from the encoded image, vector quantizationdecoding subbands in the encoded image other than a top LL subband,losslessly decoding the top LL subband, reverse wavelet transforming thetop LL subband and the vector quantization decoded subbands, andoutputting a decoded image from the decoded top LL subband and thedecoded subbands other than the decoded top LL subband.

Referring to FIG. 8, a block diagram of a method 800 of encoding,transmitting and decoding images in accordance with the presentinvention is shown. The input image is transformed into the waveletdomain using a 2-dimensional, separable octave decomposition whichgenerates a pyramid hierarchy. The top LL subband of the hierarchy iscoded losslessly. Lossless encoding methods used to code the top LLsubband may include, for example and not by way of limitation, DPCM plusHuffman encoding, DPCM plus Universal source encoding also known as Riceencoding and DPCM plus arithmetic coding. The other subbands are vectorquantized based on a zerotree insignificance prediction. Onlysignificant vectors are vector quantization (VQ) encoded. The symbols ofthe significance map are embedded in the VQ headers. Additionally, arate-distortion trade-off is applied in the vector quantization step.Decoding according to the method 800 involves the reverse of theabove-referenced procedures.

The method 800 of encoding and decoding still images by rate-distortionadaptive zerotree-based residual vector quantization of the presentinvention may be implemented in one or more integrated circuits.Referring to FIG. 9, a block diagram illustrating an image encodingcircuit 920, an image decoding circuit 930 and an image coding/decoding(or “codec”) circuit 900 are shown. The image encoding circuit 920 maybe used to encode an original image using the method 800 of FIG. 8. Anoriginal image from any source (not shown) is input into the imageencoding circuit 920. The source may be, for example and not by way oflimitation, a digital camera, a scanner, a digital data storage medium,or other digital data source. The output of the image encoding circuit920 is an image encoded according to the rate-distortion adaptivezerotree-based residual vector quantization method 800 of the presentinvention. The image encoding circuit 920 may be designed andimplemented in an integrated circuit using techniques known to one ofordinary skill in the art.

The image decoding circuit 930 may be used to decode an image encoded bythe rate-distortion adaptive zerotree-based residual vector quantizationmethod 800 of the present invention. An encoded image is input into theimage decoding circuit 930. The output of the image decoding circuit 930is a decoded image. The image decoding circuit 930 may be designed andimplemented in an integrated circuit to implement the decoding methodusing techniques known to one of ordinary skill in the art.

The method 800 of encoding and decoding still images by rate-distortionadaptive zerotree-based residual vector quantization 10, may beimplemented in a circuit card 40 embodiment of the invention. Referringto FIG. 10, a block diagram of a circuit card 40 implementing the methodof the invention is shown. Circuit card 40 may be configured to receivean original image (prior to image compression) and may output an encodedimage. Circuit card 40 may be configured to receive an encoded image andoutput a decoded image. Circuit card 40 may be configured to receiveeither an original image or an encoded image and output, respectively,either an encoded image or a decoded image. Circuit card 40 includes I/Ocircuitry 42 for communicating with external circuitry, such as forexample, a computer system (not shown). Circuit card 40 also includesimage processing circuitry 44 for encoding and decoding digital imagesaccording to the method 800 of FIG. 8. Image processing circuitry 44 maycomprise a single integrated circuit. Image processing circuitry 44 maycomprise two integrated circuits such as an image encoding circuit 920and an image decoding circuit 930 as depicted in FIG. 9. Imageprocessing circuitry 44 may include more than two discrete integratedcircuits. The image processing circuitry 44 and circuit card 40, may bedesigned to implement the rate-distortion adaptive zerotree-basedresidual vector quantization method 800 by incorporating all necessarycircuitry into, for example and not by way of limitation, an applicationspecific integrated circuit (ASIC) using techniques known to those ofordinary skill in the art.

The method of encoding and decoding still images by rate-distortionadaptive zerotree-based residual vector quantization 800 as shown inFIG. 8 may be implemented in a system for encoding, transmitting anddecoding still images. FIG. 11 is a block diagram of a system 50embodiment of the invention including the method 800 disclosed in FIG.8. System 50 includes an input device 52, a processor device 54, anoutput device 56, a storage device 58 and a memory device 60. Inputdevice 52 may be, for example and not by way of limitation, a digitalcamera, a scanner, a digital data storage medium, or other digital datasource. Output device 56 may be, for example and not by way oflimitation, a monitor, a printer. Input device 52 and output device 56may be a network interface card for communicating with external computersystems. Processor device 54 may be a general purpose microprocessor ora digital signal processor. Memory device 60 may be any form ofconventional computer memory (i.e., read only memory (ROM), dynamic readonly memory (DRAM), etc.) for storing data and/or computer programinstructions. Storage device 58 may be, for example and not by way oflimitation, a fixed hard disk, a removable media disk, e.g., floppydisk, Zip® disk, Jaz® disk, compact disk (CD) read only memory (ROM), CDrewriteable (CD-RW), magneto-optic (MO) disk, etc., or conventionalcomputer memory. System 50 may also be video system for encoding anddecoding video frames in a video system.

Another system embodiment (not shown) in accordance with the presentinvention may further include two systems 50 interconnected through acommunications channel for encoding an original image with the first oftwo systems 50, transmitting the encoded image with the first of twosystems 50 over the communications channel to a selected destination,wherein the selected destination includes a second of two systems 50,decoding the encoded image with the second of two systems 50 to providea decoded image at the selected destination.

Although this invention has been described with reference to particularembodiments, the invention is not limited to these describedembodiments. Rather, it should be understood that the embodimentsdescribed herein are merely exemplary and that a person skilled in theart may make many variations and modifications without departing fromthe spirit and scope of the invention. All such variations andmodifications are intended to be included within the scope of theinvention as defined in the appended claims.

1. A method, comprising using a system to perform the steps of:generating a hierarchy based at least on a transformation of a digitalimage comprising a low-low (LL) subband and a corresponding plurality ofthreshtrees, the threshtrees include a tree structure including aplurality of levels, each level including a plurality of vectors eachassociated with a plurality of descendent vectors of a subsequent levelup to a final level; losslessly encoding the low-low (LL) subband ofsaid hierarchy; vector quantization encoding the plurality ofthreshtrees based at least in part on a zerotree insignificanceprediction, wherein vector quantization encoding comprises selecting asignificance map of each threshtree having a minimized cost functionaccording to a rate-distortion optimization by, for each current vectorof the plurality of vectors, determining a descendent cost as a cost ofone or more descendent vectors of the current vector; determining analternate descendent cost as a cost of one or more descendent vectors ofthe current vector if all descendent vectors of the current vector areconsidered to be zerotree children; determining a significant vectorcost as a combination of the descendent cost and a cost of the currentvector; determining an isolated zero cost as a combination of thedescendent cost and a cost of coding the current vector as an isolatedzero; determining a zerotree cost as a combination of the alternatedescendant cost and a cost of coding the current vector as a zerotreeroot; symbolizing the current vector as the zerotree root if thezerotree cost is less than the isolated zero cost and less than thesignificant vector cost; symbolizing the current vector as the isolatedzero if the isolated zero cost is less than the zerotree cost and thesignificant vector cost; symbolizing the current vector as significantif the significant vector cost is less than the isolated zero cost andthe zerotree cost; and redefining the cost of all descendent vectors ofthe current vector to the alternate descendent cost if the currentvector is symbolized as the zerotree root; and generating an encodedimage based on a combination of both the encoded LL subband and theencoded plurality of threshtrees.
 2. A method according to claim 1,wherein said hierarchy comprises a pyramid hierarchy.
 3. A methodaccording to claim 1, wherein said generating the hierarchy comprisestransforming said digital image into a wavelet domain.
 4. A methodaccording to claim 1, wherein said generating the hierarchy comprisesapplying a 2-dimensional separable octave decomposition to said digitalimage.
 5. A method according to claim 1, wherein said generating thehierarchy comprises applying a Daubechies 9-7 symmetric wavelettransformation to said digital image.
 6. A method according to claim 1,wherein said generating the hierarchy comprises applying a two sixwavelet transformation to said digital image.
 7. A method according toclaim 1, wherein said generating the hierarchy comprises applying a twoten wavelet transformation to said digital image.
 8. A method accordingto claim 1, wherein said encoding the LL subband comprises applying adifferential pulse coded modulation, a Huffman coding, or combinationsthereof to said LL subband.
 9. A method according to claim 1, whereinsaid encoding the LL subband comprises applying a differential pulsecoded modulation, a universal source coding, or combinations thereof tosaid LL subband.
 10. A method according to claim 1, wherein saidencoding the LL subband comprises applying a differential pulse codedmodulation, an arithmetic coding, or combinations thereof to said LLsubband.
 11. A method according to claim 1, wherein said vectorquantization encoding comprises targeted rate control.
 12. An articlecomprising: a computer-readable medium encoded with a computer programhaving stored thereon instructions that if executed result in:generating a hierarchy based at least on a transformation of a digitalimage the hierarchy comprising low-low (LL) subband vectors and aplurality of threshtrees, the threshtrees comprising a plurality ofvectors having descendent vectors associated therewith; losslesslyencoding the LL subband vectors of said hierarchy; vector quantizationencoding the threshtrees based at least on a zerotree insignificanceprediction by, for each current vector of the plurality of vectors,determining a descendent cost as a cost of one or more descendentvectors of the current vector; determining an alternate descendent costas a cost of one or more descendent vectors of the current vector if alldescendent vectors of the current vector are considered to be zerotreechildren; determining a significant vector cost as a combination of thedescendent cost and a cost of the current vector; determining anisolated zero cost as a combination of the descendent cost and a cost ofcoding the current vector as an isolated zero; determining a zerotreecost as a combination of the alternate descendant cost and a cost ofcoding the current vector as a zerotree root; symbolizing the currentvector as the zerotree root if the zerotree cost is less than theisolated zero cost and less than the significant vector cost;symbolizing the current vector as the isolated zero if the isolated zerocost is less than the zerotree cost and the significant vector cost;symbolizing the current vector as significant if the significant vectorcost is less than the isolated zero cost and the zerotree cost; andredefining the cost of all descendent vectors of the current vector tothe alternate descendent cost if the current vector is symbolized as thezerotree root; and generating an encoded image based on a combination ofboth the encoded LL subband and the encoded threshtrees.
 13. An articleaccording to claim 12, wherein said hierarchy comprises a pyramidhierarchy.
 14. An article according to claim 13, wherein generating thehierarchy comprises applying a 2-dimensional separable octavedecomposition to said digital image.
 15. An article according to claim12, wherein generating the hierarchy comprises transforming said digitalimage into a wavelet domain.
 16. An article according to claim 12,wherein generating the hierarchy comprises applying a Daubechies 9-7symmetric wavelet transformation to said digital image.
 17. An articleaccording to claim 12, wherein generating the hierarchy comprisesapplying a two six wavelet transformation to said digital image.
 18. Anarticle according to claim 12, wherein generating the hierarchycomprises applying a two ten wavelet transformation to said digitalimage.
 19. An article according to claim 12, wherein encoding the LLsubband comprises applying a differential pulse coded modulation, aHuffman coding to said LL subband, or combinations thereof.
 20. Anarticle according to claim 12, wherein encoding the LL subband comprisesapplying a differential pulse coded modulation, a universal sourcecoding to said LL subband, or combinations thereof.
 21. An articleaccording to claim 12, wherein encoding the LL subband comprisesapplying a differential pulse coded modulation, an arithmetic coding tosaid LL subband, or combinations thereof.
 22. An article according toclaim 12, wherein said vector quantization encoding comprises targetedrate control.
 23. A method, comprising using a system to perform:generating a hierarchy based at least on a transformation of a digitalimage comprising a plurality of threshtrees, the threshtrees include atree structure including a plurality of levels, each level including aplurality of vectors each associated with a plurality of descendentvectors of a subsequent level up to a final level; vector quantizationencoding the plurality of threshtrees based at least on a zerotreeinsignificance prediction, wherein vector quantization encodingcomprises selecting a significance map of each threshtree, by, for eachof a current vector of the plurality of vectors, determining adescendent cost as a cost of one or more descendent vectors of thecurrent vector; determining an alternate descendent cost as a cost ofone or more descendent vectors of the current vector if all descendentvectors of the current vector are considered to be zerotree children;determining a significant vector cost as a combination of the descendentcost and a cost of the current vector; determining a zerotree cost as acombination of the alternate descendant cost and a cost of coding thecurrent vector as a zerotree root; symbolizing the current vector as thezerotree root if the zerotree cost is less than the significant vectorcost; symbolizing the current vector as significant if the significantvector cost is less than the zerotree cost; and redefining the cost ofall descendent vectors of the current vector to the alternate descendentcost if the current vector is symbolized as the zerotree root; andgenerating an encoded image based on the encoded plurality ofthreshtrees.
 24. The method of claim 23, wherein selecting asignificance map of each threshtree further comprises: determining anisolated zero cost as a combination of the descendent cost and a cost ofcoding the current vector as an isolated zero; and symbolizing thecurrent vector as the an isolated zero if the isolated zero cost is lessthan the zertotree cost and less than the significant vector cost.
 25. Amethod according to claim 23, wherein the hierarchy comprises a pyramidhierarchy.
 26. A method according to claim 23, wherein generating thehierarchy comprises at least one of transforming the digital image intoa wavelet domain, applying a 2-dimensional separable octavedecomposition to the digital image, applying a Daubechies 9-7 symmetricwavelet transformation to the digital image, applying a two six wavelettransformation to the digital image, and applying a two ten wavelettransformation to the digital image.
 27. A method, comprising using asystem to perform: generating a hierarchy based at least on atransformation of a digital image comprising a plurality of threshtrees,the threshtrees include a tree structure including a plurality oflevels, each level including a plurality of vectors each associated witha plurality of descendent vectors of a subsequent level up to a finallevel; vector quantization encoding the plurality of threshtrees basedat least on a zerotree insignificance prediction, wherein vectorquantization encoding comprises selecting a significance map of eachthreshtree having a reduced cost function according to a rate-distortionoptimization by, for each of a current vector of the plurality ofvectors, determining a descendent cost as a cost of one or moredescendent vectors of the current vector; determining a significantvector cost as a combination of the descendent cost and a cost of thecurrent vector; determining an isolated zero cost as a combination ofthe descendent cost and a cost of coding the current vector as anisolated zero; and symbolizing the current vector as whichever of theisolated zero and significant vector has the least cost; and generatingan encoded image based on the encoded plurality of threshtrees.
 28. Amethod according to claim 27, wherein the hierarchy comprises a pyramidhierarchy.
 29. A method according to claim 27, wherein generating thehierarchy comprises at least one of transforming the digital image intoa wavelet domain, applying a 2-dimensional separable octavedecomposition to the digital image, applying a Daubechies 9-7 symmetricwavelet transformation to the digital image, applying a two six wavelettransformation to the digital image, and applying a two ten wavelettransformation to the digital image.
 30. A method, comprising using asystem to perform: generating a hierarchy based at least on atransformation of a digital image comprising a plurality of threshtrees,the threshtrees include a tree structure including a plurality oflevels, each level including a plurality of vectors each associated witha plurality of descendent vectors of a subsequent level up to a finallevel; vector quantization encoding the plurality of threshtrees basedat least on a zerotree insignificance prediction, wherein vectorquantization encoding comprises selecting a significance map of eachthreshtree having a reduced cost function according to a rate-distortionalgorithm by, for each of a current vector of the plurality of vectors,symbolizing the current vector as either an isolated zero, zerotreeroot, or significant vector according to whichever of the isolated zeroand significant vector has the least cost according to a rate distortionalgorithm, wherein a cost of the significant vector is calculated bydetermining a descendent cost as a cost of one or more descendentvectors of the current vector; and determining a significant vector costas a combination of the descendent cost and a cost of the currentvector; and generating an encoded image based on the encoded pluralityof threshtrees.
 31. The method of claim 30, further comprisingdetermining an isolated zero cost as a combination of the descendentcost and a cost of coding the current vector as an isolated zero. 32.The method of claim 30, further comprising determining an alternatedescendent cost as a cost of one or more descendent vectors of thecurrent vector if all descendent vectors of the current vector areconsidered to be zerotree children; determining a zerotree cost as acombination of the alternate descendant cost and a cost of coding thecurrent vector as a zerotree root.